Elementary Applied Topology
The course is based on a book of Robert Ghrist,
a pleasant 250-page walk with vistas
to many applications of basic topological ideas, concepts and theorems,
that are also immensely useful in pure mathematics.
Syllabus
Manifolds. Complexes. Euler Characteristic. Homology. Sequences. Cohomology. Morse Theory. Homotopy. Sheaves. Categorification.
Bibliography
R. Ghrist, “Elementary Applied Topology”, ed. 1.0, Createspace, 2014.
ISBN 978-1502880857, Sept. 2014.
Each individual PDF chapter is available for free for personal use at author’s website https://www2.math.upenn.edu/~ghrist/notes.html
Grading
Depending on student’s track (pure of applied mathematician, or a scientist interested in applying math) the grade can be assigned after an applied project,
or a presentation, or solving some problems, or combination.
Contents
The class will choose some of the dozens topics of the book.
For every one of 10 main themes
we will discuss the basic concepts
and some application,
plus what will be required by book’s internal logic.
- Manifolds:
Configuration spaces of linkages.
Derivatives. Vector fields.
Braids and robot motion planning.
Transversality.
Signals of opportunity.
Stratified spaces.
- Complexes:
Simplicial and cell complexes.
Vietoris-Rips complexes and point clouds.
Witness complexes and landmarks.
Flag complexes and networks.
Čech complexes and random samplings.
Nerves and neurons.
Phylogenetic trees and links.
Strategy complexes and uncertainty.
Decision tasks and consensus.
Discretized graph configuration spaces.
State complexes and reconfiguration.
- Euler Characteristic:
Counting.
Curvature.
Nonvanishing vector fields.
Fixed point index.
Tame topology.
Euler calculus.
Target enumeration.
A Fubini Theorem.
Euler integral transforms.
Gaussian random fields.
- Homology:
Simplicial and cellular homology.
Homology examples.
Coefficients.
Singular homology.
Reduced homology.
Čech homology.
Relative homology.
Local homology.
Homology of a relation.
Functoriality.
Inverse kinematics.
Winding number and degree.
Fixed points and prices.
- Sequences:
Homotopy invariance.
Exact sequences.
Pairs and Mayer-Vietoris.
Equivalence of homology theories.
Cellular homology, redux.
Coverage in sensor networks.
Degree and computation.
Borsuk-Ulam theorems.
Euler characteristic.
Lefschetz index.
Nash equilibria.
The game of Hex.
Barcodes and persistent homology.
The space of natural images.
Zigzag persistence.
- Cohomology:
Duals.
Cochain complexes.
Cohomology.
Poincaré duality.
Alexander duality.
Helly’s theorem.
Numerical Euler integration.
Forms and Calculus.
De Rham cohomology.
Cup products.
Currents.
Laplacians and Hodge Theory.
Circular coordinates in data sets.
- Morse Theory:
Critical points.
Excursion sets and persistence.
Morse homology.
Definable Euler integration.
Stratified Morse theory.
Conley index.
Lefschetz index, redux.
Discrete Morse theory.
LS category.
Unimodal decomposition in statistics.
- Homotopy:
Group fundamentals.
Covering spaces.
Knot theory.
Higher homotopy groups.
Biaxial nematic liquid crystals.
Homology and homotopy.
Topological social choice.
Bundles.
Topological complexity of path planning.
Fibrations.
Homotopy type theory.
- Sheaves:
Cellular sheaves.
Examples of cellular sheaves.
Cellular sheaf cohomology.
Flow sheaves and obstructions.
Information flows and network coding.
From cellular to topological.
Operations on sheaves.
Sampling and reconstruction.
Euler integration, redux.
Cosheaves.
Bézier curves and splines.
Barcodes, redux.
- Categorification:
Categories.
Morphisms.
Functors.
Clustering functors.
Natural transformations.
Interleaving and stability in persistence.
Limits.
Colimits.
Sheaves, redux.
The genius of categorification.
“Bring out number”
Pre-requisites of point-set topology (p. 238-239) and algebra (p. 240-241)
are introduced in an appendix “Background” - 2 pages each.
Utility and natural appearance of algebraic structures
shall surface in the process of reading the book.
Related Courses.
We hope that the presented range of applications will motivate
some of the students to invest in rigorous study
of the foundations of topology (with proofs and computations):
- MAT2714. Introduction to Topology. (SD: MAT1702)
- MAT2464. Introduction to Computational Topology.
- MAT2712. Algebraic Topology I. (DF)
- MAT2713. Algebraic Topology II.
- MAT2715. Differential Topology.
- MAT2821. Differentiable Manifolds.
- MAT2923. Introduction to conservative systems.
Many of the presented concepts are also very useful
(if not indispensable) in complex analysis and algebraic geometry:
- MAT2816. Riemann surfaces II: Sheaves. Algebraic Functions. Fundamental group and singular (co)homology of compact Riemann surfaces. Monodromy. Sheaves cohomology. Harmonic forms.
- MAT2256. Introduction to Complex Geometry: Complex vector bundles, connections, curvature, Chern classes. Sheaves and cohomology. Harmonic forms, Hodge theorem and applications.
- MAT2255. Algebraic Geometry: Sheaves, Vector Bundles.